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Let $\RR \subseteq S \times T$ be a relation.

Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$, defined as:

$\RR^{-1} = \set {\tuple {t, s}: \tuple {s, t} \in \RR}$

The preimage of $\RR \subseteq S \times T$ is:

$\Preimg \RR := \RR^{-1} \sqbrk T = \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$

Also known as

Some sources, for example 1975: T.S. Blyth: Set Theory and Abstract Algebra, call this the domain of $\RR$.

However, this term is discouraged, as it is also seen used to mean the entire set $S$, including elements of that set which have no images.

Also see

Technical Note

The $\LaTeX$ code for \(\Preimg {f}\) is \Preimg {f} .

When the argument is a single character, it is usual to omit the braces:

\Preimg f